Binomial algorithms for the evaluation of options on stocks 227ple dividends; this feature is very important for the pricing of long-term options and
index options. We performed some empirical experiments and compare the results in
terms of accuracy and speed.
2 European-style options
Dividends affect option prices through their effect on the underlying stock price. In
a continuous time setting, the underlying price dynamics depends on the timing of
the dividend payment and is assumed to satisfy the following stochastic differential
equation
dSt=rStdt+σStdWt t=tD
S+tD=S−tD−DtD,(1)
whereS−tDandS+tDdenote the stock price levels right before and after the jump at time
tD, respectively. Due to this discontinuity, the solution to equation (1) is no longer
log-normal but in the form^1
St=S 0 e(r−σ(^2) / 2 )t+σWt
−DtDe(r−σ
(^2) / 2 )(t−tD)+σWt−t
DI{t≥tD}. (2)
Recently, Haug et al. [13] (henceforth HHL) derived an integral representation
formula for the fair price of a European call option on a dividend paying stock.
The basic idea is that after the dividend payment, option pricing reduces to a simple
Black-Scholes formula for a non-dividend paying stock. BeforetDone considers the
discounted expected value of the BS formula adjusted for the dividend payment. In
the geometric Brownian motion setup, the HHL formula is
CHHL(S 0 ,D,tD)=e−rtD
∫∞
dcE(Sx−D,tD)e−x(^2) / 2
√
2 π
dx, (3)
whered=log(D/S^0 )−(r−σ
(^2) / 2 )tD
σ√tD ,Sx=S^0 e
(r−σ^2 / 2 )tD+σ√tDxandc
E(Sx−D,tD)is
simply the BS formula with time to maturityT−tD. The integral representation
(3) can be considered as the exact solution to the problem of valuing a European
call option written on stock with a discrete and known dividend. Let us observe that
the well known put-call parity relationship allows the immediate calculation of the
theoretical price of a European put option with a discrete dividend.
3 American-style options
Most traded options are of American style. The effect of a discrete dividend payment
on American option prices is different than for European options. While for European-
style options the pricing problem basically arises from mis-specifying the variance
(^1) IAdenotes the indicator function ofA.